Winds around runaway stars, or in general, stars with a nonzero
relative speed with respect to the surrounding interstellar medium
(ISM), develop bullet-shaped astrospheres.

The hydrodynamic large-scale structure is sketched in
Fig.~1 and in more detail in Figre~2 the notation of which is described below.

The hypersonic stellar wind (Mach numbers \(Ma \gg 1\)) undergoes a
shock transition to subsonic velocities at the termination shock (TS)
in the inflow direction. Then a tangential discontinuity, the
astropause (AP), is formed between the ISM and the stellar wind, where
the velocity normal to it vanishes: there is no mass transport through
the AP. Other quantities such as the tangential velocity, temperature, and
density are discontinuous, while the thermal pressure is the same on
both sides. If the relative speed, or the interstellar wind speed as
seen upwind in the rest frame of the star, is supersonic in the ISM, a
bow shock (BS) exists. If the relative speed is subsonic, there will be no BS. The region
between the BS and AP is called outer astrosheath, the region between
the AP and TS the inner astrosheath. The AP around the
inflow direction at the stagnation line is sometimes called the nose,
while the region beyond the downwind TS is called the astrotail. The
latter can extend deep into the ISM. The region inside the TS is
called the inner astrosphere.

In the downwind direction, the termination shock forms a triple point,
from which the Mach disk (MD) extends down to the stagnation line; this is
the line through the stagnation point and the star. A tangential
discontinuity (TD) emerges down into the tail. A reflected shock
(not shown here) also extends from the triple point toward the TD.

This is the standard shape of an astrosphere using a single
hydrodynamic fluid \citep{Baranov-etal-1971,Pauls-etal-1995}. For the
large dimensions of O-star astrospheres, cooling operates inside
the outer astrosheath, but usually not in the inner astrosheath.
Cooling is also present beyond the model boundary (see Table~
ef{tab:2}) of the inner
astrosphere, which leads to relative sizes of these regions different
from that of pure hydrodynamic flow.

The astrosphere is always bullet-shaped, which can be seen by the
conservation of momentum:
\(\rho_{sw} v_{sw}^{2} + P_{sw} = \rho_{ISM}v_{ISM}^{2} + P_{ISM}\),
where \(\rho, v, P\) are the density, velocity, and the thermal pressure of
the stellar wind (subscript \(_{sw}\)) and the ISM (subscript \(_{ISM}\)).
The stellar wind pressure is usually neglible inside the TS. For
hypersonic flows this holds true for the thermal ISM pressure in
inflow direction, while for subsonic inflows the thermal pressure
dominates. In the tail direction, only the thermal ISM presssure is
present beyond the TS. This means that even if the inflow is subsonic,
there is a total pressure asymmetry between upwind and downwind as
long as the ISM velocity does not vanish. Thus on long time and large
spatial scales, a bullet-shaped astrosphere will develop.